cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A210516 The length-1 of the Collatz (3k+1) sequence for all odd fractions and integers.

Original entry on oeis.org

0, 1, 2, 7, 3, 3, 2, 0, 3, 6, 5, 4, 15, 7, 5, 8, 9, 3, 11, 6, 7, 16, 1, 0, 8, 2, 7, 4, 3, 4, 16, 5, 7, 25, 4, 17, 19, 5, 13, 12, 6, 7, 17, 18, 8, 6, 7, 3, 0, 3, 22, 4, 3, 8, 31, 14, 10, 6, 9, 11, 26, 12, 19, 21, 32, 10, 9, 10, 1, 31, 8, 7, 18, 2, 8, 16, 11, 76
Offset: 1

Views

Author

Michel Lagneau, Jan 26 2013

Keywords

Comments

This sequence is the unification, in the limit, of the length of Collatz sequences for all fractions whose denominator is odd, and also all integers.
The sequence A210483 gives the triangle read by rows giving trajectory of k/(2n+1) in Collatz problem, k = 1..2n, but particular attention should be paid to numbers in the triangle T(n,k) = (n-k)/(2k+1) for n = 1,2,... and k = 0..n-1.
The example shown below gives a general idea of this regular triangle. This contains all fractions whose denominator is odd and all integers. Now, from T(n,k) we could introduce a 3D triangle in order to produce a complete Collatz sequence starting from each rational T(n,k).
Remark: a(A000124(n)) = A006577(n) because the first column of this triangle generates A006577.
The triangle T(n,k) begins
1;
2, 1/3;
3, 2/3, 1/5;
4, 3/3, 2/5, 1/7;
5, 4/3, 3/5, 2/7, 1/9;
6, 5/3, 4/5, 3/7, 2/9, 1/11;
...

Examples

			The triangle of lengths begins
  0;
  1,  2;
  7,  3,  3;
  2,  0,  3,  6;
  5,  4, 15,  7,  5;
  ...
Individual numbers have the following Collatz sequences:
[1] => [0] (0 iteration);
[2  1/3] => [1, 2] because: 2 -> 1  => 1 iteration;  1/3 -> 2 -> 1 => 2 iterations;
[3  2/3  1/5] => [7, 3, 3] because: 3->10->5->16->8->4->2->1 => 7 iterations; 2/3 -> 1/3 -> 2 -> 1 => 3 iterations; 1/5 -> 8/5 -> 4/5 -> 2/5 => 3 iterations.

Links

Crossrefs

Cf. A210483, A210468, A210688.

Programs

  • Mathematica
    Collatz2[n_] := Module[{lst = NestWhileList[If[EvenQ[Numerator[#]], #/2, 3 # + 1] &, n, Unequal, All]}, If[lst[[-1]] == 1, lst = Drop[lst, -3], If[lst[[-1]] == 2, lst = Drop[lst, -2], If[lst[[-1]] == 4, lst = Drop[lst, -1], If[MemberQ[Rest[lst], lst[[-1]]], lst = Drop[lst, -1]]]]]]; t = Table[s = Collatz2[(n - k)/(2*k + 1)]; Length[s] - 1, {n, 12}, {k, 0, n - 1}]; Flatten[t] (* T. D. Noe, Jan 28 2013 *)

A210688 The length of the Collatz (3k+1) sequence for all odd fractions and integers.

Original entry on oeis.org

1, 2, 3, 8, 4, 4, 3, 1, 4, 7, 6, 5, 16, 8, 6, 9, 10, 4, 12, 7, 8, 17, 2, 1, 9, 3, 8, 5, 4, 5, 17, 6, 8, 26, 5, 18, 20, 6, 14, 13, 7, 8, 18, 19, 9, 7, 8, 4, 1, 4, 23, 5, 4, 9, 32, 15, 11, 7, 10, 12, 27, 13, 20, 22, 33, 11, 10, 11, 2, 32, 9, 8, 19, 3, 9, 17, 12
Offset: 1

Views

Author

Michel Lagneau, Jan 29 2013

Keywords

Comments

This sequence is the unification, in the limit, of the length of Collatz sequences for all fractions whose denominator is odd, and also all integers.
The sequence A210483 gives the triangle read by rows giving the trajectory of k/(2n+1) in the Collatz problem, k = 1..2n, but particular attention should be paid to numbers in the triangle T(n,k) = (n-k)/(2k+1) for n = 1,2,... and k = 0..n-1.
The example shown below gives a general idea of this regular triangle. This contains all fractions whose denominator is odd and all integers. Now, from T(n,k) we could introduce a 3D triangle in order to produce a complete Collatz sequence starting from each rational T(n,k).
The initial triangle T(n,k) begins
1;
2, 1/3;
3, 2/3, 1/5,;
4, 3/3, 2/5, 1/7;
5, 4/3, 3/5, 2/7, 1/9;
6, 5/3, 4/5, 3/7, 2/9, 1/11;
...

Examples

			The triangle of lengths begins
  1;
  2,  3;
  8,  4,  4;
  3,  1,  4,  7;
  6,  5, 16,  8,  6;
  ...
Individual numbers have the following Collatz sequences (including the first term):
[1] => [1] because: 1 -> 1 with 1 iteration;
[2 1/3] => [2, 3] because: 2 -> 2 -> 1 => 2 iterations; 1/3 -> 1/3 -> 2 -> 1 => 3 iterations;
[3 2/3 1/5] => [8, 4, 4] because: 3 -> 3->10->5->16->8->4->2->1 => 8 iterations; 2/3 -> 2/3 -> 1/3 -> 2 -> 1 => 4 iterations; 1/5 -> 1/5 -> 8/5 -> 4/5 -> 2/5 => 4 iterations.

Links

Crossrefs

Cf. A210516.

Programs

  • Mathematica
    Collatz2[n_] := Module[{lst = NestWhileList[If[EvenQ[Numerator[#]], #/2, 3 # + 1] &, n, Unequal, All]}, If[lst[[-1]] == 1, lst = Drop[lst, -3], If[lst[[-1]] == 2, lst = Drop[lst, -2], If[lst[[-1]] == 4, lst = Drop[lst, -1], If[MemberQ[Rest[lst], lst[[-1]]], lst = Drop[lst, -1]]]]]]; t = Table[s = Collatz2[(n - k)/(2*k + 1)]; Length[s] , {n, 12}, {k, 0, n - 1}]; Flatten[t] (* T. D. Noe, Jan 28 2013 *)

Formula

a(n) = A210516(n) + 1.

A224367 Triangle read by rows giving trajectory of -k/(2n+1) in Collatz problem, k = 1..2n.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 6, 9, 10, 11, 11, 13, 12, 4, 5, 1, 6, 3, 2, 4, 7, 8, 9, 8, 10, 11, 9, 13, 11, 13, 12, 14, 15, 5, 16, 16, 6, 18, 17, 20, 17, 19, 7, 4, 5, 4, 6, 1, 5, 6, 7, 7, 2, 9, 6, 8, 7, 17, 18, 9, 19, 9, 10, 20, 20, 11, 10, 22, 11, 24, 21, 23, 21, 36, 37
Offset: 0

Views

Author

Michel Lagneau, Apr 05 2013

Keywords

Comments

Extension of A210483 with negative values, and subset of A224360.

Examples

			The 2nd row [4, 5, 7, 6] gives the number of iterations of -k/5 (the first element is not counted):
   k=1 => -1/5 ->2/5 -> 1/5 -> 8/5 -> 4/5 with 4 iterations;
   k=2 => -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 5 iterations;
   k=3 => -3/5 -> -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 7 iterations;
   k=4 => -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 6 iterations.
The array starts:
  [0];
  [1, 2];
  [4, 5, 7, 6];
  [9, 10, 11, 11, 13, 12];
  [4, 5, 1, 6, 3, 2, 4, 7];
  ...

Crossrefs

Cf. A210483, A210468, A224299, A224360.

Programs

  • Mathematica
    Collatz[n_] := NestWhileList[If[EvenQ[Numerator[-#]], #/2, 3 # + 1] &, n, UnsameQ, All]; t = Join[{{0}}, Table[s = Collatz[-k/(2*n + 1)]; len = Length[s] - 2; If[s[[-1]] == 2, len = len - 1]; len, {n, 10}, {k, 2*n}]]; Flatten[t] (* program from T. D. Noe, adapted for this sequence - see A210483 *)
Showing 1-3 of 3 results.

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